Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.

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- Aug 4th 2010, 08:12 AMmikaimatrix proving problem
Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.

- Aug 4th 2010, 10:29 AMFailure
Just plug $\displaystyle \alpha u+\beta v$ for $\displaystyle x$ into the left side of $\displaystyle Ax=b$ and use the linearity of applying $\displaystyle A$ to that vector to simplify everything to just $\displaystyle b$

$\displaystyle Ax = A(\alpha u+\beta v)=\alpha (Au)+\beta (Av)=\ldots = b$